Month: October 2018

Berechnung der Konstante von Madelung

/ Danilo Roccatano / 3 Comments

Die gesamte Coulomb-Potentialenergie eines Kristalls ist die Summe der einzelnen Terme der elektrostatischen Potentialenergie

\displaystyle V_{AB} = \frac{e^2}{4\pi\epsilon_0} \frac{Z_AZ_B}{r_{AB}} \hfill (1)

zum Laden von Ionen  {q_A} e {q_B} und  getrennt nach Entfernung {r_{AB}}.

Die Summe erstreckt sich auf alle im Festkörper vorhandenen Ionenpaare für alle kristallinen Strukturen.

Die Summe konvergiert sehr langsam, weil die ersten Nachbarn des Zentralatoms einen substanziellen Beitrag zur Summe mit einem negativen Term liefern, während die benachbarten Sekunden nur mit einem etwas weicheren positiven Term beitragen, und so weiter. Auf diese Weise wird der Gesamteffekt sicherstellen, dass eine totale Initation der Anziehung zwischen Kationen und Anionen vorherrscht mit einem (negativen) Beitrag, der für die Gesamtenergie günstig ist.

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Calculus in a Nutshell: Functions and their Derivatives

/ Danilo Roccatano / Leave a comment

When I was about thirteen, the library was going to get ‘Calculus for the Practical Man.’ By this time I knew, from reading the encyclopedia, that calculus was an important and interesting subject, and I ought to learn it.  

Richard P. Feynman, from What Do You Care What Other People Think?

Introduction

Calculus is an important branch of mathematics that deals with the methods for calculating derivatives and integrals of functions and using this information to study the properties of functions. It was independently invented by I. Newton and W. Leibniz in the 18{^{th}} century and it was further developed by other great mathematicians in the centuries that follows (see Figure below).

Figure 1: Some of the great mathematician that invented the Calculus.

It comprises two areas:

  • Differential calculus {\rightarrow} It concerns the study of the rate of variation of functions.
  • Integral calculus {\rightarrow} It concern the study of the area under functions.

Depending on the nature of the functions involved in the calculations, we can further distinguish between the single- and multi-variable calculus. In this chapter, the main concepts and methods of the single-variable calculus are summarised.

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The Fourier Transform

/ Danilo Roccatano / Leave a comment

Pure mathematics is much more than an armory of tools and techniques for the applied mathematician. On the other hand, the pure mathematician has ever been grateful to applied mathematics for stimulus and inspiration. From the vibrations of the violin string they have drawn enchanting harmonies of Fourier Series, and to study the triode valve they have invented a whole theory of non-linear oscillations.

George Frederick James Temple In 100 Years of Mathematics: a Personal Viewpoint (1981).

The Fourier Transform (FT) is an integral transform, a powerful mathematical tool to map a function from its original space representation into another function space (called, in this case, the Fourier space). In the time domain, the Fourier space is the frequency and in the Cartesian domain is the so-called reciprocal space. The FT is accomplished by integrating the given function in its original space. The advantage of the FT is that in the transformed space, the properties of the original function can usually be characterised and manipulated more quickly than in the original function space. The FT function can generally be mapped back to the original function space using the inverse FT.

The FT plays an important role in pure and applied science, computer science, electronic engineering, and medicine. In this lecture, I will shortly introduce the mathematics of the FT and then show some examples of practical applications.

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The Fourier Series

/ Danilo Roccatano / Leave a comment

Pure mathematics is much more than an armory of tools and techniques for the applied mathematician. On the other hand, the pure mathematician has ever been grateful to applied mathematics for stimulus and inspiration. From the vibrations of the violin string they have drawn enchanting harmonies of Fourier Series, and to study the triode valve they have invented a whole theory of non-linear oscillations.

George Frederick James Temple In 100 Years of Mathematics: a Personal Viewpoint (1981).

Figure 1: Jean-Baptiste Joseph Fourier(source wikipedia)

The Fourier Series is a very important mathematics tool discovered by Jean-Baptiste Joseph Fourier in the 18th century. The Fourier series is used in many important areas of science and engineering. They are used to give an analytical approximate description of complex periodic function or series of data.  In this blog, I am going to give a short introduction to it.

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